I don't want to be seen to be derailing your thread but
a. perfect 12th tuning has been around since 1970 with Serge Cordier. If you're writing an academic thesis then you should be identifying why your work builds beyond his and
b. balance? What room is there for balance to make keys equal in music other than for the gamelan?
Music has been divorced from its language. The language tells us something about the music - that on account of long separation, we're missing.
What do you do with a key? :-)
What does a key really do?
It opens a door.
Where to?
A different place.
Where is the different place in an equal or quasi equal environment in which all colours are balanced to grey and all variations between black and white are carefully balanced so that they are grey?
Chromatic. What is Chromatic? What does Chromatic really mean?
In music we've forgotten that it means anything other than going up in half steps. Why the association between half steps and chromatic?
I'm sure that there are many Grey Haired colleagues on this forum who are photographers for whom George Eastman was a hero. Ekta****** . . . Koda****** were the transparency versions of what type of film? Colour! And the negative films were KodaColor . . . Chrome means Colour!
How can we see all the colours in a rainbow if they are all balanced out to be a muddy brown if you've mixed paints, or a shade of grey?
The abstraction of colour into grey is a major reason why music is not communicating emotion, because there's none to see, no contrasts beyond loud soft and fast slow, and the sustaining pedal just brings all the vibrations into a mush, leading to the "loud" pedal being exactly that, amplifying chords on every beat and being operated by pianists as a kick drum.
It's led to what is known as "vertical playing" where the music is reproduced as on a paper roll of a mechanical machine beat by beat, chord by chord, and lyricism is lost.
You tell me that I must forget perfect fifths - but that's exactly what you're finding as magic in perfect 12th tuning. On an organ that means perfect 5ths also and in fact it's 3rd harmonic alignment tuning.
That's what we need to enable resonance. But the equal spacing between semitones pits all the harmonics close together so that they interlap and beat together. This gives a mush which makes the sustain pedal unusable for its intended purpose of sustain.
We know from Chopin's correspondence that he practiced on a Pantalon. This was a simple form of piano, cheap, with wooden hammers and no dampers. How could one play complex harmonic structure on such an instrument if the result was a mush? The answer is in the tuning using the collective result to reduce the numbers of modes of vibration. When an interval is so far away from perfect ratios then the beat frequency becomes so fast that we don't hear the beats. When it's very close, then the interval is still, or very nearly so. So it resonates near pure and the ones far from the perfect don't resonate. So we remove the mush.
Attached is the beat analysis of the Kakiya Young temperament vs Equal and Kellner. In this measure ET gives 38% of coincidence of scale note to harmonics beating less than 1 beat per second, the Kakiya Young 41%, so it's an improvement and the Kellner 43%. ET produces around 15% beating 1 per second, KY 11% and Kellner 9%. 1 to 5 beats, ET 33%, KY 31% and Kellner 28%. So ET gives much more constant moving in the sound.
For a piano that will handle it, Kirnberger III gives even better results with 45% beating less than 1 per second.
If you would like to slot in the frequencies for pure 12th ET into Column AU of Sheet 1 on the spreadsheet, you'll get the extent to which 3rd harmonic tuning, or pure 12th ET improves over standard ET.
By way of my twist to Kellner, which can be applied to Kirnberger III also of which Eben Goresko is now aware and one or two temperament specialists here to whom I have disclosed on a confidential basis for experiment, I can get the number of 1 beat coincidences down from ET 15% better than standard Kellner at 9% right down to 4%. Likewise the number of 1 to 5 beat coincidences from 33% ET down beyond the 28% of Kellner down to 24%.
The results of the twisted technique on Kirnberger III will be even more so.
So it really is possible to build stillness into the tuning, reduce the mush, and restore the techniques of Chopin and Beethoven in pedalling.
The 1818 Broadwood given to Beethoven, exactly like mine of 1819
https://www.youtube.com/watch?v=GXZxR0AaXAAhas a divided sustain pedal for treble and bass. This allows special harmonic effects. This instrument, going right through to the 1859 instrument
https://www.youtube.com/watch?v=vOQ6O7PD_ycpresents next to no 5th harmonic in its sound.
Likewise an 1847 Broadwood in the Chris Maene museum
https://www.youtube.com/watch?v=mZUmZ5amfVABecause the pre-1870s instruments avoided the 5th harmonic, perfect fifth tunings, and indeed perfect 12th tunings can work extremely well on them and they can take as strong a temperament as one likes with really wide thirds. Thus on the recording of the 1859 in Kirnberger III I have difficulty in identifying the Kirnberger III from standard ET - but the resonance is changed so that power is built from the vibrations, not the brute force of the player or the extent to which the modern instrument can be abused by brute force players.
It's more musical thereby and rewards the player for listening to the sound he is making.
Logarithmic analysis of temperament is misleading and inadequate because it doesn't tell us about beating, the musical qualities and contrast of stillness and movement.
It will be very interesting if you can plug in the Perfect 12th Frequencies into Column AU on Sheet 1 and tell us the results given on the Front Page in column I.
Best wishes
David P
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David Pinnegar
East Grinstead
+44 1342 850594
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Original Message:
Sent: 01-28-2020 20:40
From: David Pinnegar
Subject: Pure 12th Equal Temperament is the Ultimate Solution for the Pythagorean Comma
I'm sorry to have to tell you that I was referring to your favourite temperament as neither beast nor fowl.
The best thirds aren't near enough pure and only four perfect fifths. It's not strong enough to make the difference to give the magic of an unequal temperament. Not strong enough to create beauty in purity and beatlessness and contrast with the stronger intervals.
I've been caught out being unable to hear Valotti as different from Equal Temperament and Vallotti with 6 perfect fifths is stronger than your favourite.
Thus my comments in that context.
Thanks. But there is a problem. In my view temperaments like this are neither beast nor fowl. With only 4 perfect fifths the difference between this and equal temperament is hardly likely to be heard. What really turns the music inside out are contrasts between beating, and beatless - clear and those so far from beatless, in other words beating so fast that they don't relate.
It's then that in the music we open up to a new set of dimensions that Equal Temperament and it's quasi equivalents can only dream of - certain - uncertain clean - dirty cold-ice - wet solid - liquid and a whole host of similar metaphors.
Furthermore, because of the increase of coincident notes and partials, the instrument resonates. Power is then achieved in the harmony, in the natural nature of the composition and its relation off vibrations, with the pedal remaining down for bars rather than beats . . . just as Beethoven and Chopin wrote for . . . rather than the brute force for which modern instruments are made to withstand and the unpleasant hitting of the keyboard.
Musicians who have experienced unequal temperament that they can hear, and which fulfills those requirements, without being nasty as some can be, go back to vanilla tuning yearning for curry again. It makes a sensitive musician so much more able to bring the emotional content of the music to life and to light.
Moving from the piano as a percussion instrument and as a gamelan to that once more which can be the reduction of the orchestra and which can sing lyrically like the flute or violin will restore interest and enthusiasm, connexion with the classical music, and spawn a whole new industry of playing, recording and instrument making as the repertoire is recorded again. Equal temperaments and their quasi modifications will spawn only more of the same. And certainly in England and France, that's downhill.
The other day I was introducing a pianist proposing to play Beethoven and Schubert on a period instrument that Vallotti was too tame and that he could go to a stronger temperament to be heard. After playing on Kellner and Kirnberger III on my instruments he remarked how it was so much more revealing for the music.
Best wishes
David P
Original Message------
David,
You said:
"In my view temperaments like this are neither beast nor fowl."
Exactly.
Balance is a rarity in the theoretical world of Unequal Temperament.
My idealised version of Thomas Young's First Temperament balances the tempering of the Fifths and the Major Thirds to the most possible extent. I have provided an extensive analysis of the mathematical design of this temperament to justify this claim.
4 Fifths and a Major Third: Reconciling the Irreconcilable Dilemma to the Most Possible Extent:
https://my.ptg.org/communities/community-home/digestviewer/viewthread?MessageKey=7d5c6389-772f-4599-bda2-3e7e2520d543&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf
You also said:
"With only 4 perfect fifths the difference between this and equal temperament is hardly likely to be heard."
You really must stop thinking about only the Pure Fifths.
The relationships between intervals need to be considered.
The balanced relationship between the Fifths and the Octaves is the key to the design of Pure 12th Equal Temperament and the balanced relationship between the Fifths and the Major Thirds is the key to the design of Roshan Kakiya's Idealised Young I.
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Roshan Kakiya
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